APPLICATIONS OF VECTORS

Scalar and Vector Quantities

Difference between Scalar and Vector Quantities

Distinguish berween scalar and vector quantities

Scalar Quantities

These are physical quantities which have magnitude only. Examples of scalar quantities include mass, length, time, area, volume, density, distance, speed, electric current and specific heat capacity

Vector Quantities

These are physical quantities which have both magnitude and direction. Examples of vector ‘quantities include displacement, velocity, acceleration, force, pressure, retardation, and momentum,

Addition of Vectors Using Graphical Method

Add vectors using graphical method Scalar physical quantities have magnitude only. Thus, they can be added, multiplied, divided, or subtracted from each other

Example 1

IF you add a volume of 40cm’ of water to a volume of 6lem’of water, then you will get 100cm'of water. Vectors ean be added, subtracted or multiplied conveniently with the help of a diagram.

Vectors Representation

A.vector quantity can be represented on paper by a direct line segment. 1, The length ofthe line seament represents the magnitude of a vector. 2, The arrow head at the end represents the direction

Methods of Vector Addition

There are two methods that are used to sum up two vectors: 1 Triangle method

• Paralletogram method,

Triangle Method

A step-by-step method for applying the head-to-tail method to determine the sum of two or more vectors is given below

• Choose a seale and indicate it on a sheet of paper. ‘The best choice of scale is one that will

result in a diagram that is as large as possible, yet fits on the sheet of paper.

• Pick a starting location and draw the first vectorto scalein the indicated direction. Label the

• Starting from where the head of the frst vector ends, draw the second vectorto scalein the

indicated direction. Label the magnitude and direction of this vector on the diagram, 4, Draw the resultant ftom the tail of the first vector to the head of the last vector, Label this vector as Resultantor simplyR,

• Using a ruler, measure the length of the resultant and determine its magnitude by converting to

real units using the scale (4.4 cm x 20 m/l cm ~ 88 m), 6 Measure the direction of the resultant using the counterclockwise convention. Resultant vector: This is the vector drawn fiom the starting point ofthe first vector to the end point of the second vector which isthe sum of two vectors R – ‘ V2 Vi Where

• Vi First veetor
• V2- Second vector
• R-Resultant vector

Example2

Suppose a man walks starting from point A, a distance of 20m due North, and then Sm due Fast. Find his new position from.

Solution

Use seale

ICM Represents Sm

Thus 20m due to North Indicates 4 em 15m due to East Indicates 3em Demonstration 15m 20m) 25m A The position of D is represented by Veetor AD of magnitude 25M or SCM at angle of 36°51 Since

• Tan Q= (Opposite /Adjacent)
• Tan Q=3em /em
• Q=Tant (34)

• Q=3s%si”

‘The Triangle and Parallelogram Laws of Forces State the triangle and parallelogram laws of forces

Triangle Law of Forces

Triangle Law of Forces siates that “IF three forces are in equilibrium and two of the forces are represented in magnitude and direction by two sides of a triangle, then the third side of the triangle represents the thd force called resultant force."

Example 3

A block is pulled by a force of 4 N acting North wards and another force 3N acting North-East Find resultant of these two forces. Demonstration c BY/3N 4N 6.5N A Seale

1Cm Represents IN

Draw a line AB of dem to the North. Then, starting from B, the top vectorofAB, draw a line BC of 3 CM at 45*East of North.

Resultant forve of two forces 3N and 4N.

Parallelogram Method

In this method, the two Vectors are drawn (usually to seale) with 2 common starting point , Ifthe lines representing the two vectors are made to be sides of s parallelogram, then the sum of the Iwo vectors will be the diagonal of the parallelogram starting from the common point The Paralelogrm Law states that “If wo vectors are epresened by the two sides given and the inclined angle between ther, then the resultant of the two vectors will be represented by the

diagonal from their common point of parallelogram formed by the two vectors”

Example 4

‘Two forces AB and AD of magnitude 40N and 60N respectively, are pulling a body on a horizontal table, If the two forces make an angle of 30° between them find the resultant force on the boty Solutvon <a

A D

Choose a cil dean cepresents1ON Draw a line AB of 4cm Draw line AD of 6em, Make an angle of 30%hetween AB and AD. Complete the parallelogram ABCD using the two sides AB and include angle 30°.

Draw the HineAC with a length 09,7 em, which is equivalent to 97 N The lincAC of the parallelogram ABCD represents the resultant force of AB and AD in idgainide aid deiod

Example

Two ropes, one 3m long and the other and 6m long, are tied to the ceiling and their free ends ate pulled by a force of JOON. Find the tension in each rope if they make an angle of 30°between them

Solution

Jom represents IN

Thus 3m ~ represent 3m

Demonstration 69N ting > 34.5N 100N By using parallelogram method

aT 34.5N Tension, determined by parallelogram method, the lengih of diagonal using scale is 8.7 om, Which represents 10ON force Thus.

‘Tension force in 3m rope is 34.5N and in 6m rope is 69N Note: Equilibrant foreesare those that act on a body at rest and counteract the force pushing or pulling the body in the opposite direction

Relative Motion

The Concept of Relative Motion

Explain the concept of relative motion Relative motion is the motion of the body relative to the moving observer. The Relative Velocity of two Bodies Cateulate the relainve velocity of two bodies Relative velocity (Vr) isthe velocity relative to the moving observer CASE 1: Ifa bus in overtaking another a passenger in the slower bus sces the overtaking bus as moving with a very small velocity CASE 2: IP the passenger was in a stationary bus, then the velocity of the overtaking bus would

appear to be greater. CASE 3: IF the observer is not stationary, then to find the velocity of body B relative to body A add velocity of B to A.

Example 6

If velocity of body B is VB and that of body A is VA, then the velocity of B with respect to A the relative velocity VBA is Given by:

VBA=VB+(-VA)

That is

VBA ~VB-VA

NOTE'The telative velocity can be oblained Graphically by applying the Triangle or parallelogram method For same direction

VBA = VB-(+VA)

=VB-VA w

For different direction

VBA = VB-(-VA)

Example 7

‘A man is swimming at 20 m/s across a river which is flowing at 10 mis. Find the resultant velocity of the man and his course if the man attempted to swim perpendicular to the water current.

Solution

Seale lemrepresents 2m/s 1CM Represents 2M/S 20m/s 22.5m/s +The length of AC is 11.25 em which is 22.5 mis moking a angle of 65°25" with the water current

• The diagonal AC represent (in magnitude and direction) the resultant velocity of the man

The Concept of Relative Motion in Daily Life

Apply the concept of relative motion in daily life Knowledge of relative motion is applied in many areas. In the Doppler effect, the received frequency depends on the relative velocity between the source and receiver. Friction force is determined by the relative motion between the surfaces in contact. Relative motions of the planets around the Sun cause the outer planets to appear as if they are moving backwards relative

to stars in universe

Resolution of Vectors

a

The Concept of Components of a Vector

Explain thc concept of camponcnisef a vector Resolution of a Vector into two Perpendicular Components

• Horizontal component
• Vertical component

Take angle OAC

Case

SinQ = EXE

Thas

EX =F SinQ

Horizontal component, FX'= FSinQ

Cos Q= Fy

Thus Fy=FCoQ Resolution of Vectors in Solving Problems Fxample Find the horizontal and vertical components of a force of 10N acting at 30° to the vertical si A 10Nsin6o? ——/ 10N —s TONcos60?

EX=FCOS 60°

os 60°F (EX)

Fy=F Sina «iy

5 ION Sin 60"